2. Determine the vertex of $f(x)=6 x^{2}+12 x+1$ using three different strategies.
Answer
Final Answer: The vertex of the function \(f(x)=6 x^{2}+12 x+1\) is \(\boxed{(-1, -5)}\).
Step 1 ：Given the function \(f(x) = 6x^{2} + 12x + 1\).
Step 2 ：The vertex of a parabola given by the equation \(f(x) = ax^{2} + bx + c\) is given by the point \(-\frac{b}{2a}, f(-\frac{b}{2a})\).
Step 3 ：Substitute the values of \(a\) and \(b\) from the given function into the formula to find the x-coordinate of the vertex: \(x = -\frac{b}{2a} = -\frac{12}{2*6} = -1\).
Step 4 ：Substitute \(x = -1\) into the function to find the y-coordinate of the vertex: \(f(-1) = 6*(-1)^{2} + 12*(-1) + 1 = -5\).
Step 5 ：Final Answer: The vertex of the function \(f(x)=6 x^{2}+12 x+1\) is \(\boxed{(-1, -5)}\).
